A100607 Concatenated primes of order 3.

223, 227, 233, 257, 277, 337, 353, 373, 523, 557, 577, 727, 733, 757, 773, 1123, 1153, 1327, 1373, 1723, 1733, 1753, 1777, 1933, 1973, 2113, 2137, 2213, 2237, 2243, 2267, 2273, 2293, 2297, 2311, 2333, 2341, 2347, 2357, 2371, 2377, 2383, 2389, 2417, 2437

Offset: 1

Author

Parthasarathy Nambi, Nov 30 2004

Keywords

• easy
• nonn
• base

Comments

This is a subset of all concatenated primes (A019549). Some of these primes have dual order - example 223. It can be viewed as order two(2 and 23) or as order three (2,2 and 3).

There are 15 such numbers less than 1000 and 202 less than 10^4. - Robert G. Wilson v, Dec 03 2004

Examples

			257 is in the sequence since it is made from three (distinct) primes.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Chris Caldwell, The First thousand primes.

Crossrefs

Cf. A019549, A100633, A383195.

Programs

Maple

filter:= proc(n) local m, i, j, ni, nj, np, n3;
 if not isprime(n) then return false fi;
 m:= ilog10(n);
 for i from 1 to m-1 do
   ni:= n mod 10^i;
   if ni < 10^(i-1) or not isprime(ni) then next fi;
   np:= (n-ni)/10^i;
   for j from 1 to m-i do
     nj:= np mod 10^j;
     if nj < 10^(j-1) then next fi;
     n3:= (np-nj)/10^j;
     if  isprime(nj) and isprime(n3) then return true fi;
 od od;
 false
end proc;
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Apr 28 2025

Mathematica

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) t = Sort[ KSubsets[ Flatten[ Table[ Prime[ Range[25]], {3}]], 3]]; lst = {}; Do[k = 1; u = Permutations[t[[n]]]; While[k < Length[u], v = FromDigits[ Flatten[ IntegerDigits /@ u[[k]]]]; If[ PrimeQ[v], AppendTo[lst, v]]; k++ ], {n, Length[t]}]; Take[ Union[lst], 45] (* Robert G. Wilson v, Dec 03 2004 *)

Formula

Each of the listed primes is made from three primes (same or different).

Extensions

Corrected and extended by Robert G. Wilson v, Dec 03 2004